Then I checked to see if I can find a connection to the ascending / descending order of the numbers, but I didn’t notice anything

Then I summed the digits of each number and noticed that sum(3rd)=sum(9th), sum(4th)=sum(10th), sum(5th)=sum(11), but all the other pairs didn’t match, so I dropped this path too

I noticed that 77=7×11, then I looked at 341. I know that only 1×1 and 3×7 give a 1 and I found out that 341 = 31×11.
Then I though to divide 923 to 13 (which in my mind was 31 inverted) => 923 = 71×13
Then I divided 1547 to 17 (71 inverted)=> 1547 = 91×17. I was surprise to see that my theory was valid so far, so I continued
608 = 32×19
2116 = 92×23
377 = 13×29
2263 = 73×31
518 = 14×37
1394 = 34×41
3182 = 74×43
1645 = 35×47
Following the above reasoning I though that the 13th number has to be a multiple of 53 => 13th = Zx53. Then I thought that 944 should be a multiple of Z inverted.
While trying to think of a solution for this number, I noticed that 7, 11, 19, 23, …, 43, 47, 53 are the prime numbers in ascending order, so 944 should be a multiple of 59 (944 = 16×59).

Knowing that the 13th number is a multiple of 53 and that 944 is a multiple of 59 => 13th number is 5035 = 95×53.

Thank you for the puzzle. It was a nice way to start my day .

http://www.testalways.com/2010/07/09/160/

for n in 77 341 923 1547 608 2116 377 2263 518 1394 3182 1645 944 4636; do echo “Testing $n…” >> out; for ((a=2;a> out; fi; done; done;

was the bash command

for n in 77 341 923 1547 608 2116 377 2263 518 1394 3182 1645 944 4636; do echo “Testing $n…” >> out; for ((a=2;a> out; fi; done; done;

The beginning of the “out” file looked like this:

Testing 77…
7
11
Testing 341…
11
31
Testing 923…
13
71
Testing 1547…
7
13
17
91
119
221

It was somewhat obvious that primes were involved. Looking at the factors laid out in this fashion actually made it simpler to recognize the reversal pattern. “Hmm, 31 and then 13; 71 and then look, 17; There must be something to this reversal.”

After going through the rest of the list and looking at the common elements from number to number in the sequence, I found this to be ascending primes, with the higher number prime reversed.

So, missing number is: 5035.

Cool puzzle! Also, very interesting to read everyone’s approaches and thoughts.

Keep these coming!

Something I want to point out that’s similar to that other numeric problem you posted a while ago: the thing that I’m really interested in, and wish I understood further, is what lies behind that initial insight that causes the problem to open (most of) its guts to you. I looked at this problem, and by the time I was aware of having thoughts about it, the little subvocalization thread over there in the back of my head was saying “look at the prime factorization of the numbers”. (a) How did it do that, and (b) how did it do that in such a way that “the rest of me” was taken by surprise? That’s the real thing, to me anyway. The rest is just working out the details. (And I’m aware of throwing in the word “just” there in a way that I shouldn’t, cf. either Gerald Weinberg or Robert Pirsig or both.)

[James' Reply: You seemed to have gone right to the factorization. I wandered through several other by-ways, first, before I remembered that Trey had once before hit me with a factors problem, so I visited easycalculation.com and played with prime factors.

I think for me, it's like this: I visit specific patterns I've seen before. If those don't work, break it down with some common forms of analysis (deltas, scatterplots, cycles, and visiting the Online Encyclopedia of Integer Sequences), looking for things that seem like repetitions or strange consistencies.]

ok, so I think this prime fatorization thing is going somewhere, let’s keep going

2263 = 31*73
518 = 2 * 7 * 39
1394 = 2 * 17 * 41
3182 = 2 * 37 * 43
1645 = 5 * 7 * 47

———-

heyyy .. 13, 17, 19, 23, 29, 31, 29, 41, 43, 47 – those are prime numbers in sequence – embarrassing that it took me this long to see it

944 = 2^4 * 59
4636 = 61 * 2^2 * 19

So the blank is going to be 53 times something. what’s the something?

11
31
71
91
2^5 = 32
2^2*23 = 92
13
73
2*7 = 14
2*17 = 34
2*37 = 74
5*7 = 35
______
2^4 = 16
2^2*19 = 76

wellll, OK, that’s a pattern on the units digit preceded by an odd number. So the blank should be either 55, 75, or 95. 75 = 3*5*5*5 which just looks different from the rest, eh? So I guess it’s between 55 = 5*11, which has the problem that the first digit is ‘5′ which doesn’t happen elsewhere in the pattern, and 95 = 5*19.

I’ve spent 10-12 minutes on this and I don’t think I’m going to get any closer than that it’s either 53 * 55 or 53 * 95 and it seems like 53 * 95 = 5035 is slightly better.

Now I’m going to read the other comments

With Regards
Mohit Verma

On the basis of my calculations the answer is 5035.

My Approach: I started with playing with numbers. I tried different mathematical operator to get some meaningful result but failed. I was gazing at numbers with hope that they would give some hint and they did. After watching the number, I decided to find the factors of each number in the series and here the answer was lying.

After factorization it looks like:
77 = 7*11,
341 =11*31,
923 = 13 * 71,
1547 = 17 * 91,
608 = 19 * 32,
2116 = 23*92,
377 = 29 * 13,
2263 = 31 * 73,
518 = 37 * 14,
1394 = 41 * 34,
3182 = 43 * 74,
1645 = 47 * 35,
x = y * z,
944 = 59 * 16,
4636 = 61 * 76

After getting this, it was easy to find y and z. y is reverse of second factor of previous number and z is the reverse of first factor of next number. Hence in the case y = 53 and z = 95 so x = 5035.

I am sure I got the right answer but still waiting for your approval.

With Regards
Mohit Verma

[James' Reply: Good job, Mohit. You are correct. Ah the power of play!]

Lots of cool puzzles in the archives of the WPC here:
http://wpc.puzzles.com/history/index.htm

[James' reply: Excellent! Yeah you were just one tiny insight short. When I solved it, I noticed the reversal once I separated the factors, but only for the first two terms, because after that there were too many factors. But when I thought to combine the factors other than the consecutive primes together that part of the pattern became clear.]